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Topological and smooth classification of Anosov maps on torus

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  15 December 2025

Ruihao Gu  and
Yi Shi
Ruihao Gu
Affiliation:
Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China ruihaogu@fudan.edu.cn
Yi Shi
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610065, China shiyi@scu.edu.cn
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Abstract

In this paper, we give a complete topological and smooth classification of non-invertible Anosov maps on torus. We show that two non-invertible Anosov maps on torus are topologically conjugate if and only if their corresponding periodic points have the same Lyapunov exponents on the stable bundles. As a corollary, if two $C^r$ non-invertible Anosov maps on torus are topologically conjugate, then the conjugacy is $C^r$-smooth along the stable foliation. Moreover, we show that the smooth conjugacy class of a non-invertible Anosov map on torus is completely determined by the Jacobians of return maps at periodic points.

Keywords

Anosov maptopological conjugacyLyapunov exponentrigidity

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 10 , October 2025 , pp. 2603 - 2653
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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